Question

If $\alpha ,\beta$ are the roots of the quadratic equation $a{x}^2+bx+c=0,a\ne 0$. Find the equation with roots ${\alpha }^2$ and ${\beta }^2.$

• A. None of the above
• B. $a(\sqrt{x}{)}^4+b(\sqrt{x}{)}^2+c=0$
• C. $(a\sqrt{x}{)}^2+(b\sqrt{x})+c=0$
• D. $a(\sqrt{x}{)}^2+b(\sqrt{x})+c=0$

Answer 1

The correct option is D $a(\sqrt{x}{)}^2+b(\sqrt{x})+c=0$

Given: $a{x}^2+bx+c=0,a\ne 0$ with roots $\alpha ,\beta$
To find: Transformed quadratic equation with roots ${\alpha }^2$ and ${\beta }^2.$

Let $t=x^2$
$\because x=\alpha \Rightarrow t={\alpha }^2$
$\&x=\beta \Rightarrow t={\beta }^2$

Replacing $x\to \sqrt{t},$ we get the transformed equation as:
$a(\sqrt{t}{)}^2+b(\sqrt{t})+c=0$
Now, in terms of variable $x,$ the equation is:
$a(\sqrt{x}{)}^2+b(\sqrt{x})+c=0$